We have seen sinusoidal images, so it follows that it ought to be possible to construct a Fourier series representation of an image. The only hard part is that images have a two-dimensional domain, and thus are finite in two distinct dimensions. The sinusoidal images that we saw correspondingly could have a vertical frequency, a horizontal frequency, or both.
Suppose that the domain of an image is [a, b] ´ [c, d] Ì Reals ´ Reals. Let pH = b - a, and pV = d- c represent the horizontal and vertical "periods" for the equivalent periodic image. For constructing a Fourier series representation, we can define the horizontal and vertical fundamental frequencies as follows:
The Fourier series representation of Image: [a, b] ´ [c, d] ® Intensity is
For convenience, we have included the constant term A0,0 in the summation, so we assume that f 0 = j 0 = 0 (recall that cos(0) = 1).